Nonexpansive Z²-subdynamics and Nivat’s Conjecture

For a finite alphabet A and η Z→ A, the Morse-Hedlund Theorem states that η is periodic if and only if there exists n∈ N such that the block complexity function P_η (n) satisfies P_η (n)≤ n, and this statement is naturally studied by analyzing the dynamics of a Z-action associated with η. In dimension two, we analyze the subdynamics of a Z²-action associated with η Z²→ A and show that if there exist n,k∈ N such that the n× k rectangular complexity Pη(n,k) satisfies Pη(n,k)≤ nk, then the periodicity of η is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n,k∈ N such that Pη(n,k)≤ nk/2, then η is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.